If a substance is an arrhenius base, is it necessarily a bronsted lowry acid

            No! If a substance is an Arrhenius Base, is it not necessarily a Bronsted Lowery Acid.

Explanation:
                   Arrhenius Base produces Hydroxyl Ions [OH⁻] when dissolved in water. While, Bronsted Lowery Acid is any specie which donates proton (H⁺). 

Example:
                           HCl  +  H₂O   →  Cl⁻  +  H₃O⁺

In this reaction HCl is a Bronsted Lowery Acid. And HCl can never produce OH⁻ ions. Hence the statement is incorrect.

                          NaOH  +  H₂O   →   Na⁺ ₍aq₎  +  OH⁻ ₍aq₎

In this reaction NaOH is ionized in water and produces OH⁻ Ions hence acting as a Arrhenius Base. And NaOH can never donate H⁺ ion so it can not act as a Bronsted Lowery Acid.